Reducibility behavior of polynomials with varying coefficients

LetK be a number field, and leth∈K[Y] be a polynomial of degreen. Fix an integer 0≤i≤n and compare the set ν of those integersa ofK such thath(Y)−aY ⁱhas a root inK with the set of those integersa, such thath(Y)−aY ⁱis reducible overK. Ifi is coprime ton, then we classify the rare cases where ν is not cofinite in . The main tools are a theorem of Siegel about integral points on algebraic curves and the theory of finite groups.     

Variational aspects of the Seiberg-Witten functional

The Seiberg-Witten equations that have recently found important applications for four-dimensional geometry are the Euler-Lagrange equations for a functional involving a connection A on a line bundleL and a section of another bundleW ⁺ constructed fromL and a spinor bundle on a given four-dimensional Riemannian manifold. We show the regularity of weak solutions and the Palais-Smale condition for this functional.     

Norm-graphs and bipartite turán numbers

For everyt>1 and positiven we construct explicit examples of graphsG with |V (G)|=n, |E(G)|c _t ·n ^2–1/t which do not contain a complete bipartite graghK _t,t !+1 This establishes the exact order of magnitude of the Turán numbers ex (n, K _t,s ) for any fixedt and allst!+1, improving over the previous probabilistic lower bounds for such pairs (t, s). The construction relies on elementary facts from commutative algebra.Research supported in part by NSF Grants DMS-8707320 and DMS-9102866.Research supported in part by Hungarian National Foundation for Scientific Research Grant     

Parameter estimation for a misspecified arma model with infinite variance innovations

We consider a general linear model , where the innovations Z_t belong to the domain of attraction of an α-stable law for α<2, so that neither Z_t nor X_t have a finite variance. We do not assume that (X_t) is a standardARMA process of the form φ(B)X_t=ϕ(B)Z_t, but we fit anARMA process of a given order to the data X₁,...,X_n by estimating the coefficients of φ and ϕ. Given that (X_t) is anARMA process, it has been proved that the Whittle estimator is a consistent estimator of the true coefficients of ϕ and φ. Moreover, it then has a heavytailed limit distribution and the rate of convergence is (n/logn)^1/α, which compares favorably with the L² situation with rate . In this note we study the limit properties of the Whittle estimator when the underlying model is not necessarily anARMA process. Under general conditions we show that the Whittle estimate converges in probability. It converges weakly to a distribution which does not have a finite moment of order a and the rate of convergence is again (n/logn)^1/α. We also give an analytic expression for the limit distribution. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

Threshold conditions for 800 nm pumping of 2 μm Tm : Ho lasers

Pumping requirements of pulsed and continuous wave (CW) 2 m Tm : Ho lasers are investigated experimentally and theoretically for excitation at 800 nm. The influence of the pulse duration on threshold is measured using a chopped Ti : Al₂O₃ laser. Based on the experimental and simulated results the spatial and temporal pump requirements in future quasi-CW high-power diode-pumped 2 m Tm : Ho lasers are determined. The experiments and the model allow the evaluation of different loss mechanisms and indicate significant 540 nm emitting loss mechanisms.